The production and diffusion of vorticity in duct flow

Abstract

Secondary flows in non-circular ducts are accompanied by a longitudinal component of vorticity. The equation of motion defining this component in a turbulent flow is composed of three terms giving the rates of production, diffusion and convection. Since the expression for production is the second derivative of Reynolds strees components, longitudinal vorticity cannot exist in laminar flow. For turbulent flow in a square duct the Reynolds stress tensor is examined in detail. Symmetry requirements alone provide relationships showing that the production is zero along all lines of symmetry. General characteristics of flow in circular pipes are sufficient to indicate where the production must be greatest. Experimental measurements verify this result and define the point density of production, diffusion and convection of vorticity. Data also indicate that the basic pattern of secondary flow is independent of Reynolds number, but that with increasing values of Reynolds number the flows penetrate the corners and approach the walls. A similar experimental investigation of a rectangular duct shows that the corner bisectors separate independent secondary flow circulation zones. Production of vorticity is again associated with the region near the bisector. However, there is some evidence that the secondary flow pattern is not so complex as inferred from the distortion of the main longitudinal flow.

Marginally turbulent flow in a square duct

“…One significant consequence of the secondary mean motion is a non negligible deformation of the primary mean velocity profile. Previous experimental measurements of the flow in a square duct (Brundrett & Baines 1964;Gessner 1973;Melling & Whitelaw 1976) as well as direct numerical simulations (Gavrilakis 1992;Huser & Biringen 1993) have provided useful reference data for the mean velocities and the Reynolds stress tensor. However, those studies were mainly focused upon the budget of the averaged flow equations, while not providing much information on the underlying physical mechanisms responsible for the formation of secondary flow.…”

Large-scale secondary structures in duct flow

“…Significant experimental contributions to the study of corner vortices arising in ducts of square cross-section have been published by Brundrett & Baines (1964) and Gessner (1973). The former observed that the vortices penetrate farther into the corners with the increase of the Reynolds number, and ascribed the secondary flows to the gradients of the normal Reynolds stresses.…”

Numerical simulation of turbulent flow through a straight square duct

“…Regardless of Reynolds number effect, the secondary flows first impinge on the corners along the bisectors and then accelerate along the walls for some distance before turning away from the walls. At the wall bisector, the present results exhibit a local maximum for the mean streamwise velocity whereas a local minimum exists in high-Reynolds-number experiments [12] (Talbe 1) that list the Reynolds numbers of the computational and experimental studies for duct flows. The occurance of a local streamwise velocity maximum at the wall bisector is a low-Reynolds-number effect [8].…”

“…Since the appearance of secondary mean motion of a turbulent flow in a straight duct flow was first measured indirectly by Nikuradse [11], a large number of experimental studies have been conducted to elucidate the dynamic response of the mean flow to the highly anisotropic turbulent field in the vicinity of an internal corner [12]. The importance of the turbulence anisotropy and the turbulent shear stress component associated with the secondary mean flow field was early recognized.…”